在创建直方图时考虑错误

Question

我有一组N个观测值在二维空间中分布为(x[i], y[i]), i=0..N个点。每个点在两个坐标（e_x[i], e_y[i], i=0..N）中都有关联的错误，并且附加了一个权重（w[i], i=0..N）。

我希望生成这些N点的二维直方图，不仅要考虑权重，还要考虑错误，这会导致每个点传播如果错误值足够大，可能在许多容器中（假设错误的标准Gaussian distribution，尽管可能会考虑其他分布）。

我看到numpy.histogram2d有一个weights参数，因此需要注意。问题是如何解释每个N观察点的错误。

有没有让我这样做的功能？我对numpy和scipy中的任何内容持开放态度。

Answer 1

根据用户1415946的评论，您可以假设每个点代表bi-variate normal distribution，其中[[e_x[i]**2,0][0,e_y[i]**2]]给出了协方差矩阵。然而，得到的分布不是正态分布 - 在运行示例之后，你会看到直方图根本不像高斯分布，而是一组代数。

要从这组分布中创建直方图，我看到的一种方法是使用numpy.random.multivariate_normal从每个点生成随机样本。请参阅下面的示例代码，其中包含一些人工数据。

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt


# This is a function I like to use for plotting histograms
def plotHistogram3d(hist, xedges, yedges):
    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')
    hist = hist.transpose()
    # Transposing is done so that bar3d x and y match hist shape correctly
    dx = np.mean(np.diff(xedges))
    dy = np.mean(np.diff(yedges))

    # Computing the number of elements
    elements = (len(xedges) - 1) * (len(yedges) - 1)
    # Generating mesh grids.
    xpos, ypos = np.meshgrid(xedges[:-1]+dx/2.0, yedges[:-1]+dy/2.0)

    # Vectorizing matrices
    xpos = xpos.flatten()
    ypos = ypos.flatten()
    zpos = np.zeros(elements)
    dx = dx * np.ones_like(zpos) * 0.5  # 0.5 factor to give room between bars.
# Use 1.0 if you want all bars 'glued' to each other
    dy = dy * np.ones_like(zpos) * 0.5
    dz = hist.flatten()

    ax.bar3d(xpos, ypos, zpos, dx, dy, dz, color='b')
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('Count')
    return

"""
INPUT DATA
"""
#                 x  y ex ey  w
data = np.array([[1, 2, 1, 1, 1],
                 [3, 0, 1, 1, 2],
                 [0, 1, 2, 1, 5],
                 [7, 7, 1, 3, 1]])

"""
Generate samples
"""
# Sample size (100 samples will be generated for each data point)
SAMPLE_SIZE = 100
# I want to fill in a table with columns [x, y, w]. Each data point generates SAMPLE_SIZE
# samples, so we have SAMPLE_SIZE * (number of data points) generated points
points = np.zeros((SAMPLE_SIZE * data.shape[0], 3))  # Initializing this matrix

for i, element in enumerate(data):  # For each row in the data set
    meanVector = element[:2]
    covarianceMatrix = np.diag(element[2:4]**2)  # Diagonal matrix with elements equal to error^2
    # For columns 0 and 1, add generated x and y samples
    points[SAMPLE_SIZE*i:SAMPLE_SIZE*(i+1), :2] = \
        np.random.multivariate_normal(meanVector, covarianceMatrix, SAMPLE_SIZE)
    # For column 2, simply copy original weight
    points[SAMPLE_SIZE*i:SAMPLE_SIZE*(i+1), 2] = element[4]  # weights

hist, xedges, yedges = np.histogram2d(points[:, 0], points[:, 1], weights=points[:, 2])
plotHistogram3d(hist, xedges, yedges)
plt.show()

结果如下：